Actually the tub and transmission don't have to come out in order to access the T-key...I guess it made it easier for the repairman but I think it's just a lot of unnecessary work. When my '55 wouldn't spin the first time around, I removed the cabinet. The second time it started "clickety-clacking", I only had to remove the front panel. Hopefully this won't happen a third time; otherwise I'm going to have to go to a machine shop to have a new one made.
Does the collar move freely? It should *immediately* drop down when you push the solenoid plunger in and bounce right up when you let go of the solenoid. If it slides down slowly, that is part of the no-spin problem and it needs lubrication. Spray PB on it first to clean it/loosen it up and then use something like 3-in-1 electric motor oil (not the "Multipurpose" type) or Zoom Spout oil.
The T-key repair can be done easily from right under the machine. In the picture you posted you can see two screws. The large screw on top is the stop bolt. Remove that one. Next, look at the side opposite the stop bolt. You should see the spin collar. Since you probably won't be able to see the T-key yet, hold the solenoid plunger in and turn the brake drum to rotate the tub . When you are able to see the T-key, let go of the solenoid and use either your hand or a small flathead screwdriver to raise the collar up. When the collar is fully raised, you can put the stop bolt back in to hold it up. Get a pair of needle-nose pliers, grab the T-key, turn it either left or right, and pull it out. Remember the direction you put it in, turn it around, re-insert it, and turn it until it is in a full vertical position and "locked" into place. Remove the stop bolt if you used it to hold the collar up, and jiggle the collar back & forth until it drops back down if it hasn't done so already. Re-insert the stop bolt, and voila, all done! Plug it in and see if it spins...it should!
One word of caution: with the T-key out, do not attempt to turn the tub or pull the trip arm!
--Austin